What Are the Minimal Conditions Required to Define a SIC POVM?

What are the minimal conditions required to deﬁne a SIC POVM?

Isabelle Jianing Geng,∗ Kimberly Golubeva,† and Gilad Gour‡ Department of Mathematics and Statistics, Institute for Quantum Science and Technology, University of Calgary, AB, Canada T2N 1N4 (Dated: July 22, 2020) Symmetric informationally complete (SIC) POVMs are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: 1) every POVM element is rank one, 2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and 3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to deﬁne a broader class of measurements which we call semi-SIC POVMs. In dimension two we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.

Introduction. Symmetric Informationally Complete where b and a are constants and d is the dimension of Positive Operator Valued Measures (SIC POVMs) are the underlying Hilbert space. objects which straddle the junction between mathematics and physics. This particular type of quantum mea- As aforementioned, in some instances a SIC POVM is surement has recently received a great deal of attention an optimal type of measurement; a consequence which in both communities because of its vast array of diverse arises from the property that a SIC POVM is comprised applications [1–23]. SICs are connected to several open of rank-one operators [25]. In [32–34], optimality was problems within the field of algebraic number theory, in- studied in the context of quantum tomography. More cluding Hilbert’s 12th problem [2–4]. Within physics, specifically, optimality refers to the minimal error in state SICs are an optimal type of quantum measurement which estimation. This concept has proven to be of signifi- have been realized experimentally [5–8], utilized in quan- cance in experimental realizations of quantum tomogra- tum information theory [9–18] and influenced the foun- phy [35]. In addition, it has been shown that when the dations of quantum mechanical theory [2, 19, 20]. De- rank one condition is relaxed, SIC POVMs exist in all spite the rapid growth of interest in these objects, a dimensions [36, 37], but in this case optimality is lost. proof of their existence in all finite dimensions–a con- The term symmetric pertains to their characterization jecture first postulated over two decades ago by Zauner– as equiangular tight frames [25] which form the vertices [24] remains elusive. Exact solutions have been found of a regular simplex in a space that contains the convex in dimensions 2 − 24, 28, 30, 31, 35, 37, 39, 43, 48, 124 [25– combinations of quantum states [38]. This condition can- 28]and numerical solutions have been found in dimen- not be relaxed as it is the integral defining characteristic sions 1 − 151 [29, 30], as well as in several other dimen- of a SIC POVM. sions up to dimension 844 [31]. 1 The value a = d is derived from the fact that any Informationally Complete (IC) POVMs posses the Pd2 POVM satisfies x=1 Ex = Id, where Id is the iden- characteristic that, when acting on a particular state, tity operator. From here, it follows that we have their statistics completely determine the quantum state. Pd2 2 x=1 Tr[Ex] = Tr[Id] = d, which implies that a = More precisely, an IC POVM is described by d pos- 1 2 Tr[E ] = . This third condition is somewhat overlooked itive semi-definite operators, {E }d , that span the x d x x=1 since it may give the impression that it follows directly d2-dimensional space of observables on a d-dimensional arXiv:2007.10483v1 [quant-ph] 20 Jul 2020 from the second condition. That is, that the value for Hilbert space H. the trace of an individual element of a SIC POVM fol- d2 lows trivially from the constant-valued Hilbert-Schmidt Definition 1. An IC POVM {Ex}x=1 is a symmetric IC-POVM (in short SIC-POVM) if it satisfies three con- inner product between any two elements of the SIC. ditions: In this paper we show that, in the two dimensional case, it is necessary to specify that the trace of an in- 2 1. Ex is rank one for all x ∈ {1, ..., d }, dividual element of a SIC POVM has a constant value. More specifically, we conclude that rather than implying 2. The Symmetry Condition; 1 Tr[Ex] = d for all x, the symmetry condition in Defini- tion1 implies that Tr[ Ex] can take at most two distinct Tr[ExEy] = b for all x 6= y , values. We use the consequences of this result to define a new class of POVMs which we refer to as semi-SIC 2 3. Tr[Ex] = a for all x ∈ {1, ..., d }. POVMs. After constructing the entire family of semi- 2

SIC POVMs in the two dimensional case, we describe a Notice that, when d = 2 the left-hand-side of (5) equals few key properties of semi-SIC POVMs in arbitrary fi- 0. That is, when d = 2, equation (5) simplifies to nite dimensions. Finally, as an application, we calculate the dual basis which enables us to represent a quantum √ state in terms of a probability vector, analogous to the 0 = (2k − 4) 1 − 12b , (6) procedure used with SIC POVMs (e.g. [38]). which implies that there are only two possibilities; Semi-SIC POVM. We begin with the definition of a 1 semi-SIC POVM. namely, either k = 2 or b = 12 . However, the latter implies that a+ = a− which means that the semi-SIC semi-SIC POVM POVM is, in fact, a SIC POVM. Hence, the possibility d2 that does not result in a SIC POVM is k = 2. Definition 2. Let {Ex}x=1 be an IC POVM acting on a Hilbert space of dimension d. Then For dimension d > 3, the left-hand-side of equation (5) d2 is equal to a positive integer. Hence, for any finite di- {Ex}x=1 is called a semi-SIC POVM if it satisfies the following conditions: mension d > 3 the value of b is given by

2 (k − d)(k + d − d2) 1. Ex is rank one for all x ∈ {1, ..., d }, b = . (7) (d2 − 1)(d2 − 2k)2 2. Tr[ExEy] = b for all x 6= y, 2 From (6) it follows that k must be no smaller than d . where b is a constant. 2 However, since b > 0, the equation above implies that k is bounded by We first show that the trace of each element of a semi- SIC POVM can take at most two distinct values. Denote 2 2 d − d < k 6 d , (8) Tr[Ex] := ax so that each element of a semi-SIC POVM can be written as Ex = ax|ψxihψx|. Then, where the upper bound follows trivially by definition. A SIC POVM corresponds to the the case k = d2 (i.e. ax = Tr[Ex] = Tr[ExId] Tr[E ] = a for all x ∈ {1, ..., d2}) which gives the value X x − = Tr[E E ] + Tr[E2] 1 x y x (1) b = d2(d+1) in (7). y6=x Construction. In the two dimensional (or qubit) case, 2 2 = (d − 1)b + ax . we are able to construct all semi-SIC POVMs up to unitary equivalence. We discover that all semi-SIC POVMs This gives us the quadratic equation can be characterized in terms of a continuous variable 2 2 ax − ax + (d − 1)b = 0 , (2) b. In the following we construct all semi-SIC POVMs in dimension two. which yields two possible values for Tr[Ex]. Namely, we 1 1 have that ax ∈ {a+, a−} where Theorem 1. Let b ∈ 16 , 12 , and define 1 ± p1 − 4b(d2 − 1) a := , (3) E1 := a−|ψ1ihψ1| ,E3 := a+|ψ3ihψ3| (9) ± 2 E2 := a−|ψ2ihψ2| ,E4 := a+|ψ4ihψ4| (10) 1 and necessarily b 6 2 . We will soon see that in 4(d −1) with a given in (3), and the 2-dimensional vectors dimensions d > 3, the parameter b can take only a few ± discrete values, whereas in dimension two, b can take a {|ψxi} given by continuous range of values. r 1 2 iθ The two distinct traces possessed by the elements Ex |ψ1i := |0i, |ψ3i := √ |0i − e |1i prompts us to introduce a new parameter k, which will 3 3 help us to determine the possible b values of a semi- r p 2 1 2 −iθ |ψ2i := r|0i + 1 − r |1i, |ψ4i := √ |0i − e |1i SIC POVM. The parameter k denotes the number of 3 3 d2 operators in the semi-SIC POVM, {Ex}x=1, with trace Tr[Ex] = a−. That is, there are k operators with trace where 2 d2 a− and d − k operators with trace a+. Since {Ex} is √ p √ ! x=1 2 b 1 − 8b − 1 − 12b a POVM, the trace of all elements must sum to d. Hence, r := √ , θ := cos−1 √ . 1 − 1 − 12b 4 b 2 d = ka− + (d − k)a+ . (4) 4 Substituting the expressions for a and a from (3) into Then, {Ex}x=1 is a semi-SIC POVM, and for any other + − 4 equation (4), we have semi-SIC POVM in dimension two, {Gx}x=1, there ex- † 4 ists a 2 × 2 unitary matrix U such that {UGxU }x=1 is d2 − 2d = (2k − d2)p1 − 4b(d2 − 1) . (5) a semi-SIC POVM of the above form. 3

Remark. The POVM constructed above is semi-SIC for Application. SIC POVMs have been used to represent all b ∈ 1 , 1 , where the case b = 1 corresponds to a quantum states as points in a probability simplex [38]. In 16 12 12 h SIC POVM. Since b ∈ 1 , 1 we must have r ∈ √1 , 1 order to emulate these results in the formalism of semi- 16 12 3 SIC POVMs, we will calculate the general form of the π π and θ ∈ 3 , 2 . dual basis of a semi-SIC POVM in arbitrary finite di- 4 mensions and use this to represent the elements of our Proof. It is straightforward to verify that {Ex}x=1 is a semi-SIC POVM. We therefore prove that all semi- two dimensional semi-SIC POVMs in terms of probabil- SIC POVMs in dimension two must take this form. ity vectors. 4 First, we calculate the dual basis of a semi-SIC POVM Let {Gx = ax|ψxihψx|}x=1 be a semi-SIC POVM with d2 a = a = a and a = a = a (recall from the ar- in arbitrary finite dimensions. Let {Ex}x=1 be a semi- 1 2 − 3 4 + d2 gument below (6) that k = 2). By applying the unitary SIC POVM and {Fy}y=1 denote its dual basis. By defini- 2 equivalence, we can assume w.l.o.g. that tion, for all y ∈ {1, ..., d }, the operators Fy must satisfy the condition p |ψ i = |0i , |ψ i = r|0i + 1 − r2 |1i and (11) 1 2 Tr[E F ] = δ . (19) q x y xy |ψ i = s |0i + 1 − s2 eiθj |1i for j = 3, 4 , (12) j j j d2 It is straightforward to check that the matrices {Fy}y=1 which satisfy the above relation are given by where sj, r ∈ [0, 1] and θj ∈ R for j = 3, 4. The condition 2 2 Tr[G1G2] = b gives  1 a+−a− 1  a2 −b Ey + (a2 −b)(1−d) S + 1−d Id if 1 6 y 6 k √ − − Fy = a2 −a2 2 b 1 E + − + T + 1 I otherwise r = √ . (13)  a2 −b y (a2 −b)(1−d) 1−d d 1 − 1 − 12b + + where The condition Tr[G1Gx] = b for x = 3, 4 gives k d2 X X 1 S := Ey and T := Ey . s3 = s4 = √ . (14) 3 y=1 y=k+1 Example 2. As an explicit example, consider the qubit The condition Tr[G2G3] = b, gives case where b = 2 . The dual basis operators are given by √ √ 25 2 p 1 − r 1 − 8b − 1 − 12b 25 5 cos(θ3) = √ = √ . (15) F = E − (E + E ) − I, 2 2r 4 b 1 2 1 2 1 2 25 5 Additionally, Tr[G2G4] = b reveals that θ3 = −θ4. F2 = E2 − (E1 + E2) − I, 2 2 (20) 25 5 It follows from (13) that r 1 corresponds to the con- F = E + (E + E ) − I, 6 3 7 3 7 3 4 dition b 1 , and r √1 corresponds to the condition > 16 > 3 25 5 b 1 . The value b = 1 is not permitted since it yields F4 = E4 + (E3 + E4) − I. 6 12 16 7 7 E1 = E2 and E3 = E4 and therefore is not information- 1 For a 2-dimensional SIC POVM the relation is more sym- ally complete. Moreover, b cannot exceed 12 since a± in (3) are not complex. That is, b ∈ 1 , 1 . metric, given as Fx = 6Ex − I2 for all x ∈ {1, 2, 3, 4}. 16 12 7 Note that, in the semi-SIC POVM, the coefficient 2 To conclude, we construct an explicit example of a breaks the symmetry in the sense that I = E1 + E2 + semi-SIC POVM in the qubit case. E3 +E4 is replaced by a weighted combination of E1 +E2

2 and E3 + E4. Example 1. Let b = 25 . Then the four elements of the corresponding semi-SIC POVM are The dual basis of the semi-SIC POVM can be used √ to derive a representation of a quantum state in terms of 2 1 0 1 1 − 2e−iθ the probability distribution associated with the outcomes E = ,E = √ , (16) 1 5 0 0 3 5 − 2eiθ 2 of the semi-SIC POVM. Specifically, since the dual basis √ 4 1 1 1 1 1 − 2eiθ {Fy}y=1 as given in (20) is a basis for the space of 2 × 2 E2 = ,E4 = √ , (17) Hermitian matrices, we can express any 2 × 2 density 5 1 1 5 − 2e−iθ 2 matrix ρ as π π where θ ∈ 3 , 2 is determined by 4 1 X cos(θ) = √ . (18) ρ = pyFy , (21) 2 2 y=1 4 where py := Tr[Eyρ] are probabilities associated with the semi-SIC measurement outcomes. Furthermore, in general, not all probability vectors T p = (p1, p2, p3, p4) will yield in (21) a positive semi- definite matrix ρ. For example, ~p = (1, 0, 0, 0) is invalid because it results in ρ = F1, which is a non-positive semi- definite matrix. We now give an example in which we characterize the set of all such probability vectors p that give rise to a 2 density matrix ρ when b = 25 . 2 Example 3. Let b = 25 . Using the dual basis calculated in Example2, ρ in Equation (21) is positive semi-definite if and only if the polynomial

4 ! X f(p) := det pyFy > 0 . (22) y=1 The polynomial f can be calculated explicitly and is FIG. 1. The probabilities which correspond to a quantum state. The green triangle is the surface with coordinates given by p1, p2, p3 where p1 + p2 + p3 = 1. The blue ellipsoid is the 8 8 9 probability distribution associated with a SIC-POVM (i.e. f(p) = − 4p2 − 4p2 − p2 − p2 + p p + 2p p b = 1 ), while the yellow ellipsoid is the distribution asso- 1 2 7 3 7 4 2 1 2 1 3 12 ciated with the semi-SIC POVM where b = 2 . Note that 9 25 + 2p1p4 + 2p2p3 + 2p2p4 + p3p4 . the areas of both distributions are located entirely above the 7 green triangle.

Since p4 = 1 − p1 − p2 − p3 the region f(p) > 0 can be characterized by of semi-SIC POVMs in Theorem1 (see an explicit ex- f(p1, p2, p3, 1 − p1 − p2 − p3) ample in Eqs. (16,17)). We generalized several defining 1 characteristics of semi-SIC POVMs in all finite dimen- = − (100p2 + 100p2 + 50p2 + 25p p + 50p p 14 1 2 3 1 2 2 3 sions, including the formulas for the values of b (see (7)) + 50p1p3 − 60p1 − 60p2 − 50p3 + 16) > 0 . and k (see (8)). Finally, we showed that the dual basis of a semi-SIC POVM can be computed and is given by This region, along with the region of a SIC-POVM, a simple formula similar to the formula of the dual of a is plotted in Figure1, which displays the area of all SIC POVM. We then used it to represent any quantum (p1, p2, p3) with p1 + p2 + p3 6 1 that corresponds to state in terms of the probability vector associated with quantum states. semi-SIC POVMs, analogous to the way it is done for In the field of quantum state tomography, the Bloch- SIC POVMs [38]. vector parametrization of a quantum state shows that Our construction of semi-SIC POVMs in two dimen- SIC POVMs can be used to form an efficient quantum to- sions reveals a parametrized family of semi-SIC POVMs 1 1 mography [39]. Additionally, it has been experimentally characterized by the parameter b ∈ 16 , 12 . This contin- implemented in [32, 35]. Inspired by this, in the appendix uous range of b is in sharp contrast to the discrete values we express the probabilities of obtaining a measurement of b valid in dimension d > 3. In particular, Eq. (7) outcome with respect to a semi-SIC POVM in terms of demonstrates that for d > 3, b can take at most d − 1 the Bloch vector representation. We expect this to be a discrete values. Thus, it may still be the case that semi- promising direction for future experiments. SIC POVMs–which are not SIC POVMs– do not exist Conclusions. In this paper, we demonstrated that in in dimension d > 3. If this is the case, it would mean dimension two, the third condition in the definition of that the conditions in Definition2 are sufficient to define a SIC POVM (see Definition1) does not follow trivially a SIC POVM. Whether semi-SIC POVMs–which are not from the second condition. In particular, we established SIC POVMs–exist in higher dimensions is left as an open that without requiring this condition, the trace of any question. given element of the IC POVM can take (at most) the GG appreciate discussions on the subject with Dante two distinct values given in (3). This prompted us in Bencivenga, Taylor Kergan, and Gaurav Saxena. The Definition2 to introduce a new class of measurements authors acknowledge support from the Natural Sciences which we referred to as semi-SIC POVMs. We then con- and Engineering Research Council of Canada (NSERC). structed the entire two dimensional one-parameter family KG acknowledge support from the Department of Math- 5 ematics and Statistics at the University of Calgary for an Adv. Sig. Proc. 2006 (2006), 10.1155/ASP/2006/85685. undergraduate research award. [24] G. Zauner, International Journal of Quantum Informa- tion 09 (2011), 10.1142/S0219749911006776. [25] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Journal of Mathematical Physics 45, 21712180 (2004). ∗ [email protected] [26] M. Appleby and I. Bengtsson, Journal of Mathematical † [email protected] Physics 60, 062203 (2019). ‡ [email protected] [27] M. Appleby, T.-Y. Chien, S. Flammia, and S. Waldron, [1] C. Fuchs, M. Hoang, and B. Stacey, Axioms 6, 21 (2017). Journal of Physics A: Mathematical and Theoretical 51, [2] M. Appleby, S. Flammia, G. McConnell, and J. Yard, 165302 (2018). Foundations of Physics 47, 10421059 (2017). [28] L. P. Hughston and S. M. Salamon, Advances in Mathe- [3] I. Bengtsson, Foundations of Physics (2020), matics 286, 1017 (2016). 10.1007/s10701-020-00341-9. [29] A. J. Scott, “Sics: Extending the list of solutions,” [4] M. Appleby, S. Flammia, G. McConnell, and J. Yard, (2017), arXiv:1703.03993 [quant-ph]. Acta Arithmetica 192, 211233 (2020). [30] A. J. Scott and M. Grassl, Journal of Mathematical [5] T. Durt, C. Kurtsiefer, A. Lamas-Linares, and Physics 51, 042203 (2010). A. Ling, Physical Review A 78 (2008), 10.1103/phys- [31] M. Grassl and A. J. Scott, Journal of Mathematical Physics 58, 122201 (2017). reva.78.042338. ˇ [6] Z. E. D. Medendorp, F. A. Torres-Ruiz, L. K. Shalm, [32]J. Reháˇcek,B.-G. Englert, and D. Kaszlikowski, Phys. G. N. M. Tabia, C. A. Fuchs, and A. M. Steinberg, Phys- Rev. A 70, 052321 (2004). ical Review A 83 (2011), 10.1103/physreva.83.051801. [33] B.-G. Englert, D. Kaszlikowski, H. K. Ng, W. K. Chua, [7] W. M. Pimenta, B. Marques, T. O. Maciel, R. O. Vianna, J. ehek, and J. Anders, “Efficient and robust quan- A. Delgado, C. Saavedra, and S. Pdua, Physical Review tum key distribution with minimal state tomography,” A 88 (2013), 10.1103/physreva.88.012112. (2004), arXiv:quant-ph/0412075 [quant-ph]. [8] J. Du, M. Sun, X. Peng, and T. Durt, Phys. Rev. A 74, [34] W. K. Wootters and B. D. Fields, Annals of Physics 191, 042341 (2006). 363 (1989). [9] C. A. Fuchs and M. Sasaki, “Squeezing quantum in- [35] N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, formation through a classical channel: Measuring the L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, Phys. ”quantumness” of a set of quantum states,” (2003), Rev. X 5, 041006 (2015). arXiv:quant-ph/0302092 [quant-ph]. [36] D. M. Appleby, Optics and Spectroscopy 103, 416 (2007). [10]J. Reháˇcek,B.-G.ˇ Englert, and D. Kaszlikowski, Phys. [37] G. Gour and A. Kalev, Journal of Physics A: Mathemat- ical and Theoretical 47, 335302 (2014). Rev. A 70, 052321 (2004). ˚ [11] B.-G. Englert, D. Kaszlikowski, H. K. Ng, W. K. Chua, [38] D. M. Appleby, A. Ericsson, and C. A. Fuchs, Founda- J. ehek, and J. Anders, “Efficient and robust quan- tions of Physics 41, 564 (2011). tum key distribution with minimal state tomography,” [39] D. Petz and L. Ruppert, Reports on Mathematical (2004), arXiv:quant-ph/0412075 [quant-ph]. Physics 69, 161 (2012). [12] A. J. Scott, Journal of Physics A: Mathematical and Gen- eral 39, 1350713530 (2006). [13] L.-M. Lai, T. Li, S.-M. Fei, and Z.-X. Wang, Quantum Appendix Inf. Process. 19, 93 (2020). [14] A. E. Rastegin, Proceedings of the Royal Society A: Let ρ be a density matrix for a qubit. Then it can be Mathematical, Physical and Engineering Sciences 471 written in terms of the Bloch parameters (rx, ry, rz) as (2015). 1 [15] M. A. Graydon and D. M. Appleby, Journal of Physics ρ = (I + r σ + r σ + r σ ) (23) A: Mathematical and Theoretical 49, 085301 (2016). 2 2 x x y y z z [16] G. N. M. Tabia and D. M. Appleby, Physical Review A where I2 is the 2×2 identity matrix and σx, σy and σz are 88 (2013), 10.1103/physreva.88.012131. three Pauli matrices. In the measurement of a semi-SIC [17] H. Zhu, Physical Review Letters 117 (2016), 10.1103/physrevlett.117.120404. POVM, by Born’s rule, qi = Tr[Eiρ] is the probability [18] C. A. Fuchs, “Qbism, the perimeter of quantum bayesian- for obtaining the outcome i ∈ [4]. The probability vector ism,” (2010), arXiv:1003.5209 [quant-ph]. q := (q1, q2, q3, q4) is in one-to-one correspondence with [19] C. A. Fuchs and B. C. Stacey, “Qbism: Quantum theory the Bloch vector r := (rx, ry, rz) via as a hero’s handbook,” (2016), arXiv:1612.07308 [quant- ph]. q1 = a−(1 + rz) [20] C. A. Fuchs and R. Schack, Reviews of Modern Physics h p i q = a 1 + 2r 1 − r2 r + 2r2 − 1 r 85, 16931715 (2013). 2 − x z [21] A. Tavakoli, I. Bengtsson, N. Gisin, and J. M. " √ ! √ ! # 2 2 2 2 1 Renes, arXiv e-prints , arXiv:2007.01007 (2020), q3 = a+ 1 − cos θ rx − sin θ ry − rz arXiv:2007.01007 [quant-ph]. 3 3 3 [22] R. Balan, B. Bodmann, and D. Edidin, Journal of " √ ! √ ! # 2 2 2 2 1 Fourier Analysis and Applications 15, 488 (2009). q = a 1 − cos θ r + sin θ r − r [23] S. Howard, R. Calderbank, and W. Moran, EURASIP J. 4 + 3 x 3 y 3 z 6 where a± , b and r are defined in the main text.